Real world example of need for quadratic equation

Question

I am (re)learning the quadratic equation. Having a concrete understanding of its purpose would really help, but I can not find any examples of a real-world scenario that requires the use of it that are more specific than "it's used by engineers" or "it's used in navigation of spacecraft." Any good real-world examples?

Answer 1

A few uses of quadratic equations, definitely not exhaustive. I'll provide one example for each domain.

• Economics: figuring out maximum profit from cost curves which are almost never linear.
• Athletics: figuring out what type of body movement/initial speed / way to throw a disk or ball will travel the furthest distance.
• War and Military: figuring out missile trajectories for offense as well as for interception.
• Space and Rockets: calculating a ton of information for the path of the rocket, fuel etc.

Answer 2

A classic example comes up when studying motion in physics, with the kinematic equation:

$$\Delta x= {v_0} t+{1\over2}a t^2$$

being a quadratic in the time variable $t$, with initial velocity $v_0$ and acceleration $a$, both held constant. Here $\Delta x$ is the displacement, or change is distance, of an object in one dimension.

If $\Delta x$ is given, along with the constant acceleration, and initial velocity, one can use the quadratic formula to solve for $t$, the time it takes the object to travel this displacement.

Ex: Suppose a car is traveling at $40$ m/s, and then starts to accelerate at a constant $1.5$ m/s$^2$. How long will it take the car to travel $10$ m?

This gives the equation:

$$10=40t+{1\over2}\cdot1.5\cdot t^2$$

Clearing decimals and fractions, and rearranging gives:

$$3t^2+160t-40=0$$

Using the quadratic formula yields:

$$t={-160\pm\sqrt{160^2-4\cdot3\cdot(-40)}\over2\cdot3}={-160\pm\sqrt{26080}\over6}\approx0.25,-53.58$$

Since time cannot be negative, we take the positive value, so that it will take the car about a quarter of a second to travel 10 m.